Abstract

Many important engineering problems, ranging from antenna design to seismic imaging, require the numerical solution of problems of time-domain propagation and scattering of acoustic, electromagnetic, elastic waves, etc. These problems present several key difficulties, including numerical dispersion, the need for computational boundary conditions, and the extensive computational cost that arises from the extremely large number of unknowns that are often required for adequate spatial resolution of the underlying three-dimensional space. In this thesis a new class of numerical methods is developed. Based on the recently introduced Fourier continuation (FC) methodology (which eliminates the Gibbs phenomenon and thus facilitates accurate Fourier expansion of nonperiodic functions), these new methods enable fast spectral solution of wave propagation problems in the time domain. In particular, unlike finite difference or finite element approaches, these methods are very nearly dispersionless---a highly desirable property indeed, which guarantees that fixed numbers of points per wavelength suffice to solve problems of arbitrarily large extent. This thesis further puts forth the mathematical and algorithmic elements necessary to produce highly scalable implementations of these algorithms in challenging parallel computing environments---such as those arising in GPU architectures---while preserving their useful properties regarding convergence and dispersion.

Additionally, this thesis develops a fast method for evaluation of computational boundary conditions which is based on Kirchhoff's integral formula in conjunction with the FC methodology and an accelerated equivalent source integration method introduced recently for solution of integral equation problems. The combination of these ideas gives rise to a physically exact radiating boundary condition that is nonlocal but fast. The only known alternatives that provide all three of these features are only applicable to a highly restrictive class of domains such as spheres or cylinders, whereas the Kirchhoff-based approach considered here only requires a bounded domain with nonvanishing thickness. As is the case with the FC scattering solvers mentioned above, the boundary-conditions algorithm is modified into a formulation that admits efficient implementation in GPU and other parallel infrastructures.

Finally, this thesis illustrates the character of the newly developed algorithms, in both GPU and parallel CPU infrastructures, with a variety of numerical examples. In particular, it is shown that the GPU implementations result in thirty- to fiftyfold speedups over the corresponding single CPU implementations. An extension of the boundary-condition algorithm, further, is demonstrated, which enables for propagation of time-domain solutions over arbitrarily large spans of empty space at essentially null computational cost. Finally, a hybridization of the FC and boundary condition algorithm is presented, which is also part of this thesis work, and which provides an interface of the newly developed algorithms with legacy finite-element representations of geometries and engineering structures. Thus, combining spectral and classical PDE solvers and propagation methods with novel GPU and parallel CPU implementations, this thesis demonstrates a computational capability that enables solution, in novel computational architectures, of some of the most challenging problems in the broad field of computational wave propagation and scattering.